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Minor Latex isssues in 2-D-1
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_problems/unit-02/D-cdf-of-a-discrete-RV/1.md

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3. Provide in bracket notion the cumulative distribution function (cdf) of X
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4. What are the probailities P(X <= 2), P(X <= 1.75) and P(X >= 2.3)
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Problem modified afterHogg, McKean and Craig - Introduction to Mathematical Statistics
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Problem modified afterHogg, McKean and Craig - Introduction to Mathematical Statistics
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---
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Let $Y$ be a bernoulli random variable defined such that each flip
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$$
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P(Y = y) =
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\begin{cases}
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0.5, &y = 0 \\
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0.5, &y = 1 \\
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0.5, &y = 0 \\\\
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0.5, &y = 1 \\\\
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0, & \text{otherwise}
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\end{cases}
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$$
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That is we sum all the mass from $\( -\infty \)$ to desired value of $x$ and recall that this cumulative probability quantity only changes at values of $X$ for which there is positive mass. Since as we recall probability is only in the range of 0 to 1, and recalling that cdf increases at increasing supp values, we have...
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$$
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\(F_{X}(x) = \left\\{
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\begin{array}{ll}
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0 & \text{if } x < 0 \\
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0.0625 & \text{if } 0 \leq x < 1 \\
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0.3125 & \text{if } 1 \leq x < 2 \\
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0.6875 & \text{if } 2 \leq x < 3 \\
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0.9375 & \text{if } 3 \leq x < 4 \\
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F_{X}(x) =
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\begin{cases}
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0 & \text{if } x < 0 \\\\
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0.0625 & \text{if } 0 \leq x < 1 \\\\
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0.3125 & \text{if } 1 \leq x < 2 \\\\
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0.6875 & \text{if } 2 \leq x < 3 \\\\
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0.9375 & \text{if } 3 \leq x < 4 \\\\
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1 & \text{if } 4 \leq x \\
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\end{array}
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\right. \)
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\end{cases}
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$$
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For part 4 we can just read off of the cdf in bracket notation

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